Class Session 08: Heat-bath algorithm for the Ising model

Introduction

We discuss a simple algorithm for the Ising model at equilibrium with a bath at inverse temperature β (see section 5.2.2 in SMAC). The system is described by a set of N spins {σ1,...,σN} each of them taking the value +1 or –1. The energy of the system is proportional to the Boltzman weight of energy

$E[\{\sigma\}]=-\displaystyle \sum_{\langle ij\rangle} \sigma_i \sigma_j$

where the sum is over neighboring sites.

The heat bath algorithm

During the lecture we have discussed the Metropolis algorithm for this system. Rather than flipping a spin at a random site, we now thermalize this spin with its local environment: the state of a spin after a flip does not depend on its state before the flip, but only on its molecular field h (defined by the sum of the spins of its neighbors).
Show that, in the presence of a molecular field h at site k, the spins point up and down with probabilities:

$\displaystyle \pi_h^+ = \frac 1{1+e^{-2\beta h}} \qquad \qquad \pi_h^- = \frac 1{1+e^{+2\beta h}}$

Those define the transition probabilities at each step of the heat bath algorithm (for a uniformly chosen spin to become respectively up or down, irrespective of its initial state). Show that detailed balance is verified with respect to the Boltzmann-Gibbs weight at inverse temperature β. The following program implements this algorithm in two dimension on a square lattice of lenght L with periodic boundary conditions:

import random,math,pylab,numpy
L = 100
Nstep = 100000
beta = 0.3
neighbors = [(0,-1),(0,1),(-1,0),(1,0)]
#initialisation
s = [[ 1 for i in range(L)] for j in range(L)]
energy = -2. * L**2
toten = 0.
#evolution
for step in range(Nstep):
    kx = random.randrange(L)
    ky = random.randrange(L)
    h = sum( s[(kx+dx)%L][(ky+dy)%L] for (dx,dy) in neighbors)
    olds = s[kx][ky]
    Upsilon = random.random()
    s[kx][ky] = -1
    if Upsilon < 1./(1+math.exp(-2*beta*h)): s[kx][ky] = 1
    if olds != s[kx][ky]: energy-=2*h*s[kx][ky]
    toten += energy
print toten / Nstep / L**2
print sum(sum(s[i][j] for i in range(L)) for j in range(L) )/L**2.
pylab.matshow(numpy.array(s))
pylab.show()

Half order and coupling algorithms

Figure 1. Half order in the Ising model.

Let us first introduce the concept of half order for the configurations in the Ising model (see figure 1). We define that, for a site, an up spin is larger than a down spin, and a whole configuration {σ1,...,σN} of spins is larger than an other configuration {σ'1,...,σ'N} if the spins on each sites k satisfy σk≥σ'k. This defines an half-order among the configurations (half, because not all configurations can be compared)..

Show that the half order is preserved at each iteration of the heat bath algorithm.
In the forward coupling, show that it is sufficient to focus on the coupling of two particular configurations. Which ones?

The same remark applies for the backward coupling, and allows to draw a perfect sample. Here is an implementation:

import random,math,pylab,numpy
L=10
beta=0.01
neighbors=[(0,-1),(0,1),(-1,0),(1,0)]
#initialisation
splus0=[[ 1 for i in range(L)] for j in range(L)]
sminus0=[[ -1 for i in range(L)] for j in range(L)]
splus,sminus=splus0[:],sminus0[:]
changes=[]
#evolution
while splus!=sminus :
    splus,sminus=splus0[:],sminus0[:]
    kx=random.randrange(L)
    ky=random.randrange(L)
    Upsilon=random.random()
    changes.insert(0,[kx,ky,Upsilon])
    for [kx,ky,Upsilon] in changes:
        for s in (splus,sminus):
            h=sum( s[(kx+dx)%L][(ky+dy)%L] for (dx,dy) in neighbors)
            s[kx][ky]=-1
            if Upsilon<1./(1+math.exp(-2*beta*h)): s[kx][ky]=1
print "here is a perfect sample", splus
print "steps to find it", len(changes)
pylab.matshow(numpy.array(splus))
pylab.show()

What is the advantage of this method compared to the naive implementation of the backward coupling?

[Print this page]

Heat-bath algorithm for the Ising model

Table of Contents

Introduction

The heat bath algorithm

Half order and coupling algorithms